We analyze a nonlinear pricing model with limited information. Each buyer can purchase a large variety, d, of goods. His preference for each good is represented by a scalar and his preference over d goods is represented by a d-dimensional vector. The type space of each buyer is given by a compact subset of Rd+ with a continuum of possible types. By contrast, the seller is limited to offer a finite number M of d-dimensional choices.
We provide necessary conditions that the optimal finite menu of the social welfare maximizing problem has to satisfy. We establish an underlying connection to the theory of quantization and provide an estimate of the welfare loss resulting from the usage of the d-dimensional M-class menu. We show that the welfare loss converges to zero at a rate proportional to d/M2/d.
We show that in higher dimensions, a significant reduction in the welfare loss arises from an optimal partition of the d-dimensional type space that takes advantage of the correlation among the d parameters.
(with Stephen Morris) “Equilibrium robustness in informational variables is critical, if one wants to use results from the mechanism design literature in real life applications. The papers included in the Bergemann and Morris book describe state of the art progress in this direction of research. The book is an excellent resource for established game theorists, who want to learn more about this subject; and for PhD students, who look for exciting problems to investigate.” ─ Ehud Kalai, Kellogg School of Management, Northwestern University
“This book collects together a series of papers on mechanism design written by Dirk Bergemann and Stephen Morris. It is their response to the challenge set by Robert Wilson in his eponymous doctrine: Only by repeated weakening of common knowledge assumptions will the theory approximate reality. Many scholars responded by arguing for solution concepts robust to the beliefs of the agents. The approach taken by Bergemann and Morris was radically different. They hitched their wagon to Harsany’s observation that relaxing the common knowledge assumption was equivalent to enlarging the type space. Then, they proceed to develop the properties of mechanisms that would emerge. For this reason, this collection is essential reading for any student interested in taking up the challenge of the Wilson doctrine. The introduction by itself is worth the price of admission!” ─ Rakesh Vohra, Kellogg School of Management, Northwestern University
“Mechanism design has been one of the great successes of economic theory in the last 30 years. Robust mechanism design, the study of optimal mechanisms in settings where the designer has less information about the beliefs of the agents, is the natural next step in the evolution of this field. Bergemann and Morris are two of the leading figures in developing this new theory, and this book combines many of their papers with an excellent introduction that overviews the field and explains how their papers fit together. Highly recommended to all students and practitioners of economic theory, and essential reading for would-be mechanism designers.” ─ Drew Fudenberg, Department of Economics, Harvard University
“The question of the design of institutions has been at the center of some of the most important economic theory in the past four decades. Bergemann and Morris have made seminal contributions to the understanding of how uncertainty can and should be incorporated into mechanism design, and this volume reproduces a collection of their most important work in the area. The volume will be an important reference for those working in the area and those who wish to apply the ideas in economic models.” ─ Andrew Postlewaite, Department of Economics, University of Pennsylvania
We consider the efficient allocation of a single good with interdependent values in a quasi-linear environment. We present an approach to modelling interdependent preferences distinguishing between “payoff types” and “belief types” and report a characterization of when the efficient allocation can be partially Bayesian implemented on a finite type space. The characterization can be used to unify a number of sufficient conditions for efficient partial implementation in this classical auction setting.
We report how a canonical language for discussing interdependent types — developed in a more general setting by Bergemann, Morris and Takahashi (2011) — applies in this setting and note by example that this canonical language will not allow us to distinguish some types in the payoff type — belief type language.
We consider the optimal design of flexible use in a digital-rights-management policy for a digital good subject to piracy. Consumers can acquire the digital good either as a licensed product or as an unlicensed copy. The ease of access to unlicensed copies is increasing in the flexibility accorded to licensed copies. The content provider has to trade off consumers’ valuation of a licensed copy against the sales lost to piracy.
We enrich the basic model by introducing a “secure platform” that is required to use the digital good. We show that the platform allows for the socially optimal provision of flexibility for the digital good but only if both are sold by an integrated firm.
We define a notion of correlated equilibrium for games with incomplete information in a general setting with finite players, finite actions, and finite states, which we call Bayes correlated equilibrium. The set of Bayes correlated equilibria of a fixed incomplete information game equals the set of probability distributions over actions, states and types that might arise in any Bayes Nash equilibrium where players observed additional information. We show that more information always shrinks the set of Bayes correlated equilibria.
The set of outcomes that can arise in Bayes Nash equilibria of an incomplete information game where players may or may not have access to more private information is characterized and shown to be equivalent to the set of an incomplete information version of correlated equilibrium, which we call Bayes correlated equilibrium. We describe a partial order on many player information structures — which we call individual sufficiency — under which more information shrinks the set of Bayes correlated equilibria. We discuss the relation of the solution concept to alternative definitions of correlated equilibrium in incomplete information games and of the partial order on information structures to others, including Blackwell’s for the single player case.
We analyze games of incomplete information and offer equilibrium predictions which are valid for, and in this sense robust to, all possible private information structures that the agents may have. We completely characterize the set of Bayes correlated equilibria in a class of games with quadratic payoffs and normally distributed uncertainty in terms of restrictions on the first and second moments of the equilibrium action-state distribution. We derive exact bounds on how prior knowledge about the private information refines the set of equilibrium predictions.
We consider information sharing among firms under demand uncertainty and find newly optimal information policies via the Bayes correlated equilibria. Finally, we reverse the perspective and investigate the identification problem under concerns for robustness to private information. The presence of private information leads to set rather than point identification of the structural parameters of the game.
We analyze games of incomplete information and offer equilibrium predictions which are valid for all possible private information structures that the agents may have. Our characterization of these robust predictions relies on an epistemic result which establishes a relationship between the set of Bayes Nash equilibria and the set of Bayes correlated equilibria.
We completely characterize the set of Bayes correlated equilibria in a class of games with quadratic payoffs and normally distributed uncertainty in terms of restrictions on the first and second moments of the equilibrium action-state distribution. We derive exact bounds on how prior information of the analyst refines the set of equilibrium distribution. As an application, we obtain new results regarding the optimal information sharing policy of firms under demand uncertainty.
Finally, we reverse the perspective and investigate the identification problem under concerns for robustness to private information. We show how the presence of private information leads to partial rather than complete identification of the structural parameters of the game. As a prominent example we analyze the canonical problem of demand and supply identification.
We analyze games of incomplete information and offer equilibrium predictions which are valid for, and in this sense robust to, all possible private information structures that the agents may have. We completely characterize the set of Bayes correlated equilibria in a class of games with quadratic payoffs and normally distributed uncertainty in terms of restrictions on the first and second moments of the equilibrium action-state distribution. We derive exact bounds on how prior knowledge about the private information refines the set of equilibrium predictions.
We consider information sharing among firms under demand uncertainty and find newly optimal information policies via the Bayes correlated equilibria. Finally, we reverse the perspective and investigate the identification problem under concerns for robustness to private information. The presence of private information leads to set rather than point identification of the structural parameters of the game.
We analyze games of incomplete information and offer equilibrium predictions which are valid for, and in this sense robust to, all possible private information structures that the agents may have. The set of outcomes that can arise in equilibrium for some information structure is equal to the set of Bayes correlated equilibria. We completely characterize the set of Bayes correlated equilibria in a class of games with quadratic payoffs and normally distributed uncertainty in terms of restrictions on the first and second moments of the equilibrium action-state distribution. We derive exact bounds on how prior knowledge about the private information refines the set of equilibrium predictions.
We consider information sharing among firms under demand uncertainty and find new optimal information policies via the Bayes correlated equilibria. We also reverse the perspective and investigate the identification problem under concerns for robustness to private information. The presence of private information leads to set rather than point identification of the structural parameters of the game.
This essay is the introduction for a collection of papers by the two of us on “Robust Mechanism Design” to be published by World Scientific Publishing. The appendix of this essay lists the chapters of the book.
The objective of this introductory essay is to provide the reader with an overview of the research agenda pursued in the collected papers. The introduction selectively presents the main results of the papers, and attempts to illustrate many of them in terms of a common and canonical example, the single unit auction with interdependent values.
In addition, we include an extended discussion about the role of alternative assumptions about type spaces in our work and the literature, in order to explain the common logic of the informational robustness approach that unifies the work in this volume.